Optimal Quantized Compressed Sensing via Projected Gradient Descent

TL;DR

This paper develops a unified analysis of quantized compressed sensing using projected gradient descent, achieving optimal error rates for various models including 1-bit and multi-bit scenarios.

Contribution

It introduces conditions under which PGD attains near-optimal recovery error rates in quantized compressed sensing, extending to multi-bit and dithered models.

Findings

PGD achieves the optimal rate O(k/mL) for sparse signals.

Error rates match or improve existing sharp bounds.

Unified treatment applies to multiple quantization schemes.

Abstract

This paper provides a unified treatment to the recovery of structured signals living in a star-shaped set from general quantized measurements $\mathcal{Q}(\mathbf{A}\mathbf{x}-\mathbf{\tau})$, where $\mathbf{A}$ is a sensing matrix, $\mathbf{\tau}$ is a vector of (possibly random) quantization thresholds, and $\mathcal{Q}$ denotes an $L$-level quantizer. The ideal estimator with consistent quantized measurements is optimal in some important instances but typically infeasible to compute. To this end, we study the projected gradient descent (PGD) algorithm with respect to the one-sided $\ell_1$-loss and identify the conditions under which PGD achieves the same error rate, up to logarithmic factors. These conditions include estimates of the separation probability, small-ball probability and some moment bounds that are easy to validate. For multi-bit case, we also develop a complementary approach based on product embedding to show global convergence. When applied to popular models such as 1-bit compressed sensing with Gaussian $\mathbf{A}$ and zero $\mathbf{\tau}$ and the dithered 1-bit/multi-bit models with sub-Gaussian $\mathbf{A}$ and uniform dither $\mathbf{\tau}$, our unified treatment yields error rates that improve on or match the sharpest results in all instances. Particularly, PGD achieves the information-theoretic optimal rate $\tilde{O}(\frac{k}{mL})$ for recovering $k$-sparse signals, and the rate $\tilde{O}((\frac{k}{mL})^{1/3})$ for effectively sparse signals. For 1-bit compressed sensing of sparse signals, our result recovers the optimality of normalized binary iterative hard thresholding (NBIHT) that was proved very recently.